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Finite generation of split F-regular monoid algebras

Rankeya Datta, Karl Schwede, Kevin Tucker

TL;DR

We establish that for a submonoid $S$ of a free abelian group of finite rank and a field $k$ of prime characteristic, split-$F$-regularity of the monoid algebra $k[S]$ forces finite generation of $k[S]$ (equivalently, $S$ is finitely generated). The approach translates the problem to convex-cone geometry via $oldsymbol{\sigma_S}$, uses Carathéodory’s theorem, and applies Diophantine-approximation (BCHM-density) to show the cone is rational polyhedral, yielding finite generation. We further develop the equivalence of split-$F$-regularity and $F$-pure-regularity in this setting and derive independence of the ground field; these results enable a robust framework for valuations: $ ext{gr}_ u(R)$ is finitely generated under split-$F$-regularity for maximal rank Abhyankar valuations. Collectively, these findings support the expectation that split-$F$-regular rings in function fields are Noetherian and have broad applications to Cox rings and symbolic Rees algebras in birational geometry and valuation theory.

Abstract

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or equivalently, that $S$ is a finitely generated monoid. Split $F$-regular rings are possibly non-Noetherian or non-$F$-finite rings that satisfy the defining property of strongly $F$-regular rings from the theories of tight closure and $F$-singularities. Our finite generation result provides evidence in favor of the conjecture that split $F$-regular rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry.

Finite generation of split F-regular monoid algebras

TL;DR

We establish that for a submonoid of a free abelian group of finite rank and a field of prime characteristic, split--regularity of the monoid algebra forces finite generation of (equivalently, is finitely generated). The approach translates the problem to convex-cone geometry via , uses Carathéodory’s theorem, and applies Diophantine-approximation (BCHM-density) to show the cone is rational polyhedral, yielding finite generation. We further develop the equivalence of split--regularity and -pure-regularity in this setting and derive independence of the ground field; these results enable a robust framework for valuations: is finitely generated under split--regularity for maximal rank Abhyankar valuations. Collectively, these findings support the expectation that split--regular rings in function fields are Noetherian and have broad applications to Cox rings and symbolic Rees algebras in birational geometry and valuation theory.

Abstract

Let be a submonoid of a free Abelian group of finite rank. We show that if is a field of prime characteristic such that the monoid -algebra is split -regular, then is a finitely generated -algebra, or equivalently, that is a finitely generated monoid. Split -regular rings are possibly non-Noetherian or non--finite rings that satisfy the defining property of strongly -regular rings from the theories of tight closure and -singularities. Our finite generation result provides evidence in favor of the conjecture that split -regular rings in function fields over have to be Noetherian. The key tool is Diophantine approximation from convex geometry.
Paper Structure (22 sections, 54 theorems, 159 equations, 2 figures)

This paper contains 22 sections, 54 theorems, 159 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof sketch
  3. $F$-pure-regularity
  4. An application to graded rings associated with valuations
  5. The geometry of convex cones
  6. Convex cones
  7. Dual cones and duality
  8. Strong convexity
  9. Faces of cones
  10. Relative interiors of cones and their duals
  11. Extremal subcones and lineality spaces
  12. Diophantine approximation
  13. Finite generation of split-$F$-regular monoid algebras
  14. Background on split-$F$-regularity and $F$-pure-regularity
  15. From monoids to convex cones
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose $M = \mathbb{Z}^d$, and $\sigma \subseteq M \otimes_{\mathbb Z} \mathbb{R}$ is a full-dimensional convex cone containing the origin. Suppose that $k$ is a field of characteristic $p > 0$. If $k[\sigma \cap M]$ is split-$F$-regular, then $\sigma$ is a closed cone generated by finitely many ra

Figures (2)

  • Figure 1.1: A convex cone with accumulating extremal rays, intersected with an affine hyperplane.
  • Figure 1.2: A convex cone generated by $\sigma \cap H$ in the affine hyperplane $H$, with new chosen origin.

Theorems & Definitions (132)

  • Conjecture
  • Theorem : \ref{['thm:SFR-cone-finite-generation']}
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 2.1.1: Carathéodory
  • proof
  • Lemma 3
  • ...and 122 more